Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $q = \dfrac{4}{16t^2 + 72t} \div \dfrac{-2}{5t(2t + 9)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{4}{16t^2 + 72t} \times \dfrac{5t(2t + 9)}{-2} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ 4 \times 5t(2t + 9) } { (16t^2 + 72t) \times -2 } $ $ q = \dfrac {4 \times 5t(2t + 9)} {-2 \times 8t(2t + 9)} $ $ q = \dfrac{20t(2t + 9)}{-16t(2t + 9)} $ We can cancel the $2t + 9$ so long as $2t + 9 \neq 0$ Therefore $t \neq -\dfrac{9}{2}$ $q = \dfrac{20t \cancel{(2t + 9})}{-16t \cancel{(2t + 9)}} = -\dfrac{20t}{16t} = -\dfrac{5}{4} $